Complex aerospace structures typically include unknown states, parameters, or inputs. The unknown parameters may be due to changes in the structure that are not captured by the mathematical model assumed. These models are often reduced order models (ROM) that have simplified physics or have been obtained through data-driven techniques, such as trained neural networks. In this paper, we evaluate two data assimilation techniques to perform parameter estimation of dynamical systems by leveraging measured responses to correct process model predictions. We study two different noise models: discontinuous and continuous Gaussian noises. We use ensemble Kalman filter and Kalman-Bucy filter techniques on representative structures, such as the slender flat beam with nonlinear features to illustrate how this approach could be applied to more complex structures.
The efficient condition assessment of engineered systems requires the coupling of high fidelity models with data extracted from the state of the system ‘as-is’. In enabling this task, this paper implements a parametric Model Order Reduction (pMOR) scheme for nonlinear structural dynamics, and the particular case of material nonlinearity. A physics-based parametric representation is developed, incorporating dependencies on system properties and/or excitation characteristics. The pMOR formulation relies on use of a Proper Orthogonal Decomposition applied to a series of snapshots of the nonlinear dynamic response. A new approach to manifold interpolation is proposed, with interpolation taking place on the reduced coefficient matrix mapping local bases to a global one. We demonstrate the performance of this approach firstly on the simple example of a shear-frame structure, and secondly on the more complex 3D numerical case study of an wind turbine tower under a ground motion excitation. Parametric dependence pertains to structural properties, as well as the temporal and spectral characteristics of the applied excitation. The developed parametric Reduced Order Model (pROM) can be exploited for a number of tasks including monitoring and diagnostics, control of vibrating structures, and residual life estimation of critical components.
Neural networks (NNs) are known as universal function approximators and can interpolate nonlinear functions between observed data points. However, when the target domain for deployment shifts from the training domain and NNs must extrapolate, the results are notoriously poor. Prior work Martinez et al. (2019) has shown that NN uncertainty estimates can be used to correct binary predictions in shifted domains without retraining the model. We hypothesize that this approach can be extended to correct real-valued time series predictions. As an exemplar, we consider two mechanical systems with nonlinear dynamics. The first system consists of a spring-mass system where the stiffness changes abruptly, and the second is a real experimental system with a frictional joint that is an open challenge for structural dynamicists to model efficiently. Our experiments will test whether 1) NN uncertainty estimates can identify when the input domain has shifted from the training domain and 2) whether the information used to calculate uncertainty estimates can be used to correct the NN’s time series predictions. While the method as proposed did not significantly improve predictions, our results did show potential for modifications that could improve models’ predictions and play a role in structural health monitoring systems that directly impact public safety.
Soft ferromagnetic alloys are often utilized in electromagnetic applications due to their desirable magnetic properties. In support of these applications, the ferromagnetic alloys are also required to bear mechanical load under various loading and environmental conditions. In this study, a Fe–49Co–2V alloy was dynamically characterized in tension with a Kolsky tension bar and a Drop–Hopkinson bar at various strain rates and temperatures. Dynamic tensile stress–strain curves of the Fe–49Co–2V alloy were obtained at strain rates ranging from 40 to 230 s−1 and temperatures from − 100 to 100 °C. All dynamic tensile stress–strain curves exhibited an initial linear elastic response to an upper yield followed by Lüders band response and then a nearly linear work-hardening behavior. The yield strength of this material was found to be sensitive to both strain rate and temperature, whereas the hardening rate was independent of strain rate or temperature. The Fe–49Co–2V alloy exhibited a feature of brittle fracture in tension under dynamic loading with no necking being observed.
Fe-Co-2V is a soft ferromagnetic alloy used in electromagnetic applications due to excellent magnetic properties. However, the discontinuous yielding (Luders bands), grain-size-dependent properties (Hall-Petch behavior), and the degree of order/disorder in the Fe-Co-2V alloy makes it difficult to predict the mechanical performance, particularly in abnormal environments such as elevated strain rates and high/low temperatures. Thus, experimental characterization of the high strain rate properties of the Fe-Co-2V alloy is desired, which are used for material model development in numerical simulations. In this study, the high rate tensile response of Fe-Co-2V is investigated with a pulse-shaped Kolsky tension bar over a wide range of strain rates and temperatures. Effects of temperature and strain rate on yield stress, ultimate stress, and ductility are discussed.
Cylindrical dog-bone (or dumbbell) shaped samples have become a common design for dynamic tensile tests of ductile materials with a Kolsky tension bar. When a direct measurement of displacement between the bar ends is used to calculate the specimen strain, the actual strain in the specimen gage section is overestimated due to strain in the specimen shoulder and needs to be corrected. The currently available correction method works well for elastic-perfectly plastic materials but may not be applicable to materials that exhibit significant work-hardening behavior. In this study, we developed a new specimen strain correction method for materials possessing an elastic-plastic with linear work-hardening stress–strain response. A Kolsky tension bar test of a Fe-49Co-2V alloy (known by trade names Hiperco and Permendur) was used to demonstrate the new specimen strain correction method. This new correction method was also used to correct specimen strains in Kolsky tension bar experiments on two other materials: 4140 alloy, and 304L-VAR stainless steel, which had different work-hardening behavior.
In computational structural dynamics problems, the ability to calibrate numerical models to physical test data often depends on determining the correct constraints within a structure with mechanical interfaces. These interfaces are defined as the locations within a built-up assembly where two or more disjointed structures are connected. In reality, the normal and tangential forces arising from friction and contact, respectively, are the only means of transferring loads between structures. In linear structural dynamics, a typical modeling approach is to linearize the interface using springs and dampers to connect the disjoint structures, then tune the coefficients to obtain sufficient accuracy between numerically predicted and experimentally measured results. This work explores the use of a numerical inverse method to predict the area of the contact patch located within a bolted interface by defining multi-point constraints. The presented model updating procedure assigns contact definitions (fully stuck, slipping, or no contact) in a finite element model of a jointed structure as a function of contact pressure computed from a nonlinear static analysis. The contact definitions are adjusted until the computed modes agree with experimental test data. The methodology is demonstrated on a C-shape beam system with two bolted interfaces, and the calibrated model predicts modal frequencies with <3% total error summed across the first six elastic modes.
Proceedings of ISMA 2018 - International Conference on Noise and Vibration Engineering and USD 2018 - International Conference on Uncertainty in Structural Dynamics
Advanced friction models are often mathematically defined as nonlinear differential equations or complicated algebraic operations acting in single degree-of-freedom systems; however, such simplified conditions are not relevant to most design applications. As a result, current designers of practical structures typically simplify friction modeling to classical, Coulomb-like descriptions. In order to be viable for design purposes, friction models must be applicable to realistic structures and available in standard commercial codes. The goal of this work is to implement several different friction models into the commercial code, Abaqus, as user-defined contact models and to explore their properties in a dynamic simulation. A verification problem of interest to the joints community is utilized to evaluate efficacy. Several output quantities of the model will be presented and discussed, including frictional energy dissipation, amplitude, and frequency. The selected results are comparable to commonly observed experimental phenomena in mechanics of jointed structures.
Proceedings of ISMA 2018 - International Conference on Noise and Vibration Engineering and USD 2018 - International Conference on Uncertainty in Structural Dynamics
Roettgen, D.; Jones, E.; Brink, Adam R.; Reu, P.; Rohe, D.; Wagner, J.
Mechanical and aerospace structures often contain nonlinearities arising from frictional contact at joints. In order to calibrate response predicting models, these nonlinearities must be experimentally quantified to provide information about the type and strength of nonlinearity. For amplitude dependent nonlinearities, such as frictional contact, the nonlinear response is obtained by gathering ring down accelerometer data from impact testing with varying amplitudes. These accelerations are then spatially filtered to obtain a single degree-of-freedom response, which is used to identify a pseudo-modal model. This work examines a small test article with embedded frictional nonlinearities, in which accelerometers cannot be placed. Using a laser doppler vibrometer (LDV) to determine linear mode shapes and digital image correlation (DIC) to obtain response data during various amplitude hammer strikes, the nonlinearity in this small system is quantified.
Proceedings of ISMA 2018 - International Conference on Noise and Vibration Engineering and USD 2018 - International Conference on Uncertainty in Structural Dynamics
Complex mechanical structures are often subjected to random vibration environments. One strategy to analyze these nonlinear structures numerically is to use finite element analysis with an explicit solver to resolve interactions in the time domain. However, this approach is impractical because the solver is conditionally stable and requires thousands of iterations to resolve the contact algorithms. As a result, only short runs can be performed practically because of the extremely long runtime needed to obtain sufficient sampling for long-time statistics. The proposed approach uses a machine learning algorithm known as the Long Short-Term Memory (LSTM) network to model the response of the nonlinear system to random input. The LSTM extends the capability of the explicit solver approach by taking short samples and extending them to arbitrarily long signals. The efficient LSTM algorithm enables the capability to perform Monte Carlo simulations to quantify model-form and aleatoric uncertainty due to the random input.
Component mode synthesis (CMS) is a widely employed model reduction technique used to reduce the computational cost associated with the dynamic analysis of complex engineering structures. To generate CMS models, specifically the formulation of Craig and Bampton, both normal fixed-interface modes and constraint modes of the component’s structure are calculated. These modes are used in conjunction with the component level mass and stiffness matrices to generate reduced mass and stiffness matrices used in the final analyses. For some component models, the most computationally expensive part of this procedure is calculating the component normal modes information. Several different approaches are utilized to investigate the sensitivity of system level responses to variations in several aspects of the CMS models. One approach evaluates changes due to modifications of the reduced mass and stiffness matrices assuming that the mode shapes do not change. The second approach assumes that the mode shapes change but the reduced mass and stiffness matrices do not change. An example is presented to show the influence of these two approaches.
Structural dynamics models with localized nonlinearities can be reduced using Hurty/Craig-Bampton component mode synthesis methods. The interior degrees-of-freedom of the linear subcomponents are reduced with a set of dynamic fixedinterface modes while the static constraint modes preserve the physical coordinates at which the nonlinear restoring forces are applied. For finite element models with a highly refined mesh at the boundary, a secondary modal analysis can be performed to reduce the interface down to a truncated set of local-level characteristic constraint modes. In this research, the cost savings and accuracy of the interface reduction technique are evaluated on a simple example problem involving two elastic blocks coming into contact.
This work describes the energy dissipation arising from microslip for an elastic shell incorporating shear and longitudinal deformation resting on a rough-rigid foundation. This phenomenon is investigated using finite element (FE) analysis and nonlinear geometrically exact shell theory. Both approaches illustrate the effect of shear within the shell and observe a reduction in the energy dissipated from microslip as compared to a similar system neglecting shear deformation. In particular, it is found that the shear deformation allows for load to be transmitted beyond the region of slip so that the entire interface contributes to the load carrying capability of the shell. The energy dissipation resulting from the shell model is shown to agree well with that arising from the FE model, and this representation can be used as a basis for reduced order models that capture the microslip phenomenon.
Proceedings of ISMA 2016 - International Conference on Noise and Vibration Engineering and USD2016 - International Conference on Uncertainty in Structural Dynamics
We propose the use of the Primary Complex Mode Indicator Function (PCMIF), calculated from acceleration frequency response functions, as a response comparison metric to analyze unit-to-unit variability. The PCMIF has an advantage over the traditional dynamic representations of mode shapes, frequencies and damping in that it removes the user and algorithmic error that may be associated with those extractions. In addition, it is customizable according to the interest level. If consistent sets of acceleration measurements from chosen drive points can be acquired from multiple hardware units, the comparison of each unit's PCMIF metric can provide insight. In addition to measured PCMIF, finite element models of the system can predict variability in PCMIF response using known variability of hardware configurations, and this can be compared with experimental PCMIF data. This comparison allows meaningful unit-to-unit comparison even if components and/or system geometries differ from one system to the next.
One of the more severe environments for a store on an aircraft is during the ejection of the store. During this environment it is not possible to instrument all component responses, and it is also likely that some instruments may fail during the environment testing. This work provides a method for developing these responses from failed gages and uninstrumented locations. First, the forces observed by the store during the environment are reconstructed. A simple sampling method is used to reconstruct these forces given various parameters. Then, these forces are applied to a model to generate the component responses. Validation is performed on this methodology.